Snapback Repellers in Chaos Theory and their Application in Ecology

• Main Authors: Chen, Hsun-Hui, 陳勛暉
• Other Authors: Chang, Shu-Ming
• Format: Others
• Language: zh-TW
• Published: 2010
• Online Access: http://ndltd.ncl.edu.tw/handle/12384229665052138639

碩士 === 國立交通大學 === 應用數學系所 === 98 === This work briefly reviews the history of chaos theory and elucidates the relationship among chaos, Lyapunov exponent, topological entropy, strange attractor, snapback repeller, and Sarkovskii's theorem, connecting them to each other using a relational graph. Mathematical and computer-assisted tools can be used to determine whether maps or systems are chaotic by finding a quantity or sometimes identifying the existence of a property. In ecology, Satake's generalized resource budget model that modified from Isagi's resource budget model, Satake and Iwasa proved by computing the positive Lyapunov exponent that if the depletion coefficient k is greater than one, then the system is chaotic. However, a positive Lyapunov exponent means only sensitivity in Devaney's chaos. Therefore, this work presents mathematical views and a numerical analysis on Satake's model, using the "snapback repeller method" to prove that the model is chaotic in Devaney's sens (involving transitivity, density, and sensitivity). Moreover, this work also overcomes the omission of Satake's paper (Satake & Iwasa, 2000) when the depletion coefficient k is a positive integer. Furthermore, the end of this work investigates the difference between odd depletion coefficients and even depletion coefficients.